Velocity statistics inside coherent vortices generated by the inverse cascade of 2-D turbulence

Kolokolov, I. V.; Лебедев, В. В.

doi:10.1017/jfm.2016.699

Cited by 33 publications

(15 citation statements)

References 26 publications

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“…The zeroth mode, m = 0, deserves a separate consideration (see also 10,33 ). As was noted before, and can be deduced from the (φ, φ) component of the steady state equation…”

Section: Fluctuationsmentioning

confidence: 99%

The culmination of an inverse cascade: Mean flow and fluctuations

Frishman

2017

Physics of Fluids

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Two dimensional turbulence has a remarkable tendency to self-organize into large, coherent structures, forming a mean flow. The purpose of this paper is to elucidate how these structures are sustained, and what determines them and the fluctuations around them. A recent theory for the mean flow will be reviewed. The theory assumes turbulence is excited by a forcing supported on small scales, and uses a linear shear model to relate the turbulent momentum flux to the mean shear rate. Extending the theory, it will be shown here that the relation between the momentum flux and mean shear is valid, and the momentum flux is non-zero, for both an isotropic and an anisotropic forcing, independent of the dissipation mechanism at small scales. This conclusion requires taking into account that the linear shear model is an approximation to the real system. The proportionality between the momentum flux and the inverse of the shear can then be inferred most simply on dimensional grounds. Moreover, for a homogeneous pumping, the proportionality constant can be determined by symmetry considerations, recovering the result of the original theory. The regime of applicability of the theory, its compatibility with observations from simulations, a formula for the momentum flux for an inhomogeneous pumping, and results for the statistics of fluctuations, will also be discussed.

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“…The zeroth mode, m = 0, deserves a separate consideration (see also 10,33 ). As was noted before, and can be deduced from the (φ, φ) component of the steady state equation…”

Section: Fluctuationsmentioning

confidence: 99%

The culmination of an inverse cascade: Mean flow and fluctuations

Frishman

2017

Physics of Fluids

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“…In the same work an interval of separations from the vortex center was found where the flat velocity profile is realized and some theoretical arguments toward the flat profile were formulated. In our paper [20] we demonstrated that the flat profile corresponds to the passive (quasilinear) regime of turbulent fluctuations, and in [21] we calculated the structure function of the velocity fluctuations. In all the works the model of the pumping short correlated in time was used.…”

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confidence: 90%

“…Therefore a question appears about an internal structure of the vortices. For a short-correlated in time pumping, the problem was investigated in the works [19][20][21]. Let us analyze the structure of the coherent vortex in the static case.…”

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confidence: 99%

Profile of coherent vortices in two-dimensional turbulence

Kolokolov

Лебедев

2015

Jetp Lett.

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We examine the velocity profile of coherent vortices appearing as a consequence of the inverse cascade of two-dimensional turbulence in a finite box in the case of static pumping. We demonstrate that in the passive regime the flat velocity profile is realized, as in the case of pumping short correlated in time. However, in the static case the energy flux to large scales is dependent on the system parameters. We demonstrate that it is proportional to f 4/3 where f is the characteristic force exciting turbulence.Turbulence is a chaotic state of fluid motions realized at large Reynolds numbers Re, see, e.g., [1]. In a number of cases turbulence appears to be effectively two-dimensional [2]. Already first theoretical works [3][4][5] devoted to two-dimensional turbulence reveal its principal difference from three-dimensional one. The difference is related to existence of two quadratic quantities (energy and enstrophy) conserved by two-dimensional Euler equation. That leads to two different cascades produced by the non-linear interaction: enstrophy is carried from the pumping scale to smaller scales (direct cascade) whereas energy is carried to larger scales (inverse cascade). The enstrophy is dissipated due to viscosity at scales smaller than the pumping length and the energy is dissipated due to bottom friction at scales larger than the pumping length.Statistical properties of the velocity fluctuations in the inverse cascade were investigated both experimentally [6] and numerically [7]. Results of the works are in a good agreement with the analytical theory developed for an unbound system [8]. Particularly, the normal Kolmogorov scaling is observed in the inverse cascade [2]. The normal scaling in the inverse cascade of twodimensional turbulence is in contrast with the anomalous scaling observed in three-dimensional turbulence [9]. Some theoretical arguments in favor of the normal scaling in the inverse cascade were presented in the work [10], where the feature was related to a leading role of the converging Lagrangian trajectories in the inverse cascade.In an unbound two-dimensional system the inverse cascade is terminated at a scale L α , which is determined by a balance between the energy flux ǫ (per unit mass) toward large scales and the bottom friction. Using the Kolmogorov estimate (ǫL α ) 1/3 for the velocity at the scale L α , one obtains from the balance L α = ǫ 1/2 α −3/2 , where α is the bottom friction coefficient. If the box size L is smaller than L α , then the energy carried by the inverse cascade to scales of the order of the box size L starts to accumulate there. The scenario is realized if the pumping scale l is much less than the box size L, l ≪ L, otherwise there is no space for the inverse cascade carrying the energy to the scale L.The accumulation of the energy at the scale L leads to appearing an intensive large-scale motion including big coherent vortices. The tendency towards the formation of the vortices was indicated already in the first works devoted to two-dimensional turbulence, both exp...

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“…The average turbulent energy profile.-Symmetry considerations imply that the magnitude of the diagonal terms of the Reynolds tensor, u 2 and v 2 , as well as the mechanism that dictates them, are different than those for the momentum flux [10,11]. We therefore take a different approach than [17], where the turbulent energy is computed within the same framework used for the momentum flux. Instead, we will show that u 2 and v 2 are determined by the zero modes of the advection equation for two-point correlation functions.…”

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Turbulence Statistics in a Two-Dimensional Vortex Condensate

Frishman

Herbert

2018

Phys. Rev. Lett.

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Disentangling the evolution of a coherent mean-flow and turbulent fluctuations, interacting through the nonlinearity of the Navier-Stokes equations, is a central issue in fluid mechanics. It affects a wide range of flows, such as planetary atmospheres, plasmas, or wall-bounded flows, and hampers turbulence models. We consider the special case of a two-dimensional flow in a periodic box, for which the mean flow, a pair of box-size vortices called "condensate," emerges from turbulence. As was recently shown, a perturbative closure describes correctly the condensate when turbulence is excited at small scales. In this context, we obtain explicit results for the statistics of turbulence, encoded in the Reynolds stress tensor. We demonstrate that the two components of the Reynolds stress, the momentum flux and the turbulent energy, are determined by different mechanisms. It was suggested previously that the momentum flux is fixed by a balance between forcing and mean-flow advection: using unprecedently long numerical simulations, we provide the first direct evidence supporting this prediction. By contrast, combining analytical computations with numerical simulations, we show that the turbulent energy is determined only by mean-flow advection and obtain for the first time a formula describing its profile in the vortex.

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Velocity statistics inside coherent vortices generated by the inverse cascade of 2-D turbulence

Cited by 33 publications

References 26 publications

The culmination of an inverse cascade: Mean flow and fluctuations

The culmination of an inverse cascade: Mean flow and fluctuations

Profile of coherent vortices in two-dimensional turbulence

Turbulence Statistics in a Two-Dimensional Vortex Condensate

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